Exploring the Origin of the Born Rule in Quantum Experimental Phenomena

[ThomasT]

With regard to quantum experimental phenomena, might not one think of the movement (light and sound indicators and subsequent recorded data streams) of some piece of macroscopic instrumentation as the results of measurements of the energy of wavelike disturbances within some volume of space during some intervals of time. That is, data streams in quantum experimental setups are generated by the energy imparted from quantum systems to macroscopic instruments.

The Born rule says that the probability of a quantum system to produce a changes in macroscopic instrumentation is directly proportional to the square of the amplitude of the quantum system.

With respect to classical systems, the energy transported by a wave (and imparted to obstacles in the wave's path) is approximately equal to the square of the amplitude of the wave.

Is this where the Born rule came from?

 

[Hurkyl]

My favorite justification of the Born rule is this:

In a certain sense, the only really relevant property of a quantum state $\psi$ is that it determines an 'expected value' $\langle T \rangle_\psi$ for any observable $T$. Then, you invoke a deep theorem of functional analysis that says that if you define states this way, then for any particular state $\psi$ you can find a (good) Hilbert space representation in which $\psi$ 'factors' into a bra and a ket satisfying $\langle T \rangle_\psi = \langle \psi | T | \psi \rangle$, for some ket $|\psi \rangle$ for all observables T. Conclusion: the Born rule is really just a mathematical artifact of the way Hilbert spaces can naturally be used to represent states. Of course, the discovery process went in the opposite direction...

 

[ps2138]

I just did a quick search for a question I had and figured I could put it in this thread because I'm guessing its somewhat related. I apologize ahead of time if the answer could have been unearthed with a little more effort on my part looking into my statistics or qm textbooks (this question actually popped into my head as I was reviewing Griffiths' 1st chapter) or if its frankly just a silly question.

I understand that with respect to the Born interpretation of the wave function that the actual probability of finding a particle (the probability of a quantum system to produce a change in this case) is given by the product of the wave function with its complex conjugate. This will yield a non-negative real probability for any normalizable function. However, I've been searching for the proof to this interpretation to understand why this is true mathematically. If this is an approximation why shouldn't the probability be proportional to any non-trig even function or specifically to say something like wave function⁴ or even an approximation expansion as opposed to the wave function squared?

I've done a cursory search on this forum and found this https://www.physicsforums.com/showthread.php?t=237940&highlight=born+statistical+interpretation" but it looks like the last comment just answers with an experimental justification and the post previous to that doesn't really address my question.

Even if there isn't a proof (for whatever reason) I'd greatly appreciate a reply to my question [even if its just to point me to a helpful link or to tell me where I'm getting mixed up]---I understand there is a good chance that my question is just a fundamental misunderstanding of the application of statistics to the probability equation (though I did take a second to look at other distribution functions) but I know that leaving the issue unresolved will eat at it me even if it’s just an unnecessary diversion while I continue to review the book.

Thanks again.

 

[reilly]

With regard to quantum experimental phenomena, might not one think of the movement (light and sound indicators and subsequent recorded data streams) of some piece of macroscopic instrumentation as the results of measurements of the energy of wavelike disturbances within some volume of space during some intervals of time. That is, data streams in quantum experimental setups are generated by the energy imparted from quantum systems to macroscopic instruments.

The Born rule says that the probability of a quantum system to produce a changes in macroscopic instrumentation is directly proportional to the square of the amplitude of the quantum system.

With respect to classical systems, the energy transported by a wave (and imparted to obstacles in the wave's path) is approximately equal to the square of the amplitude of the wave.

Is this where the Born rule came from?

Its a good bet that, indeed, Born was in part motivated by the content of Poynting's Thrm, and the general importance of wave intensity in E&M and other wave-dominated fields. As a student, I had the good fortune to talk with a few physicists who learned and worked during Born's time, and learned my basic QM from one of them-- J.H. Van Vleck. I got the strong sense that Born's ideas were related to the usual interpretation of wave intensity, including the E&M approach to scattering -- see the classical derivation of the Rutherford scattering cross section, for example.

Lots of the early work in QM was motivated by physical reasoning.The more abstract approach with Hilbert Spaces and state vectors, vs coordinate or momentum based wave functions really took off with the publication of Dirac's book in the 1930s.

And, note that Born's interpretation is just that; it is justified by its consistent utility over nearly a century of application.
Regards,
Reilly Atkinson

 

[Maaneli]

<< With respect to classical systems, the energy transported by a wave (and imparted to obstacles in the wave's path) is approximately equal to the square of the amplitude of the wave.

Is this where the Born rule came from? >>

Schroedinger first tried to interpret rho = |psi|^2 as a continuous matter density, much like the classical intensity of a wave. But that wasn't consistent with the empirical observation of particles. Born then came along and showed that a probability interpretation of rho is self-consistent if one assumes it is a probability density distribution for discrete events like the "measurement" (the term was not even defined back then). In particular, one can construct an operator algebra that allows one to correctly compute the expectation values of various observables using rho as the probability measure.

This is only a postulate that seems to nevertheless work extremely well. But it is not a dynamical, physical derivation of the Born rule. However, in the de Broglie-Bohm (deBB) pilot wave theory, one can actually derive the Born rule (rather than postulating it) in two different ways. One can show that rho is the measure of greatest typicality for point particles trajectories in 3-space, and that any hypothetical initial particle distribution where rho \= |psi|^2 will converge to rho = |psi|^2 and stay there for all t since it is the only equivariant measure. Or, one can show from a subquantum H-theorem that if an ensemble of deBB particles are initially distributed as rho \= |psi|^2, the mixing of the particles in nonequilibrium will relax very quickly to statistical equilibrium given by rho = |psi|^2, and will stay there because it is the only equivariant measure. Notice that both of these arguments are parallel to the justifications given by Boltzmann for the 2nd law of thermodynamics. The first is in fact an application of Boltzmann's "statistical argument" (also known as the typicality argument), and the second is just an application of Boltzmann's H-theorem. This is one reason why pilot wave theory is a more general theory than textbook QM. See the following papers:

Quantum Equilibrium and the Origin of Absolute Uncertainty
Authors: Dürr, Detlef; Goldstein, Sheldon; Zanghí, Nino
http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Aquant-ph%2F0308039

Dynamical Origin of Quantum Probabilities
Antony Valentini and Hans Westman
http://eprintweb.org/S/authors/All/va/Valentini/12

Hidden Variables, Statistical Mechanics and the Early Universe
Antony Valentini
http://eprintweb.org/S/authors/All/va/A_Valentini/14

 

[ThomasT]

Thanks to all for your comments, links, and references.